An Accurate Numerical Method and Algorithm for Constructing Solutions of Chaotic Systems

Abstract

In various fields of natural science, the chaotic systems of differential equations are considered more than 50 years. The correct prediction of the behaviour of solutions of dynamical model equations is important in understanding of evolution process and reduce uncertainty. However, often used numerical methods are unable to do it on large time segments. In this article, the author considers the modern numerical method and algorithm for constructing solutions of chaotic systems on the example of tumor growth model. Also a modification of Benettin's algorithm presents for calculation of Lyapunov exponents.

On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

“…Next, consider the algorithm [20] for the numerical construction of an approximate solution of a dynamical system in forward and backward time: Set X(0) ∈ B a for the system (2) and value the number way that determines the direction in time: way = 1 is gone forward in time, way = −1 is gone backward in time; 4.…”

“…Next, consider the algorithm [20] for the numerical construction of an approximate solution of a dynamical system in forward and backward time:…”

“…In [20], a modification of the algorithm for calculating the Lyapunov exponents for a model describing the growth of a cancerous tumor was considered. Next, we will consider a generalization of this algorithm for dynamical systems with a quadratic right-hand side of the n-th order.…”

“…This article considers a fourth-order dissipative system [22] in which, in the opinion of its authors, there is hyperchaos. The goals of the research are as follows: (1) to find the Poincaré recurrences in the attractor of this system for the approximate solutions obtained by the FGBFI method and to analyze the statistics of return times; and (2) to apply the modified Benettin's algorithm [20] to refine the computation of the Lyapunov exponents.…”

“…In general, the problem of numerical modeling with control of the accuracy of obtained solution and the choice of a platform for computer implementation is relevant today [19], since small errors introduced at each integration step cause exponential divergence of close trajectories. More recently, the numerical FGBFI method (Firmly Grounded Backward-Forward Integration) has been proposed in [3,4,20], based on the power series method for dynamical systems with quadratic nonlinearities. The main advantages of this method are as follows:…”